Hi Bogdan,
This is a lot of separate questions in one post, so I can only respond in a general way.
Yes, you can apply effects to an image taken with the camera. But you may have some issues doing it during the camera operation, unless you use the “camera to texture” feature which is available only for iOS, and then depending on how intensive the filter effect, you may have performance issues. For example, if you try to execute a composite “gaussian blur + bloom + sepia” filter on a live camera texture, the performance will not be acceptable.
You can effectively cut, crop, and draw on the photo, but these are completely separate topics which would require you research the methods on your own.
Posting to Instagram is not supported, but you can share the photo in other ways depending on which platform:
https://coronalabs.com/blog/2015/03/17/tutorial-utilizing-the-activity-popup-plugin-ios/
http://docs.coronalabs.com/plugin/CoronaProvider_native_popup_social/index.html
Take care,
Brent